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Version 1996-05-04 10 (9P) Paper Sh:566, version 1996-05-04 10. See https://shelah.logic.at/papers/566/ for possible updates. A COMPLETE BOOLEAN ALGEBRA THAT HAS NO PROPER ATOMLESS COMPLETE SUBLAGEBRA Thomas Jech and Saharon Shelah The Pennsylvania State University, The Hebrew University and Rutgers University Abstract. There exists a complete atomless Boolean algebra that has no proper atomless complete subalgebra. An atomless complete Boolean algebra B is called simple [5] if it has no atomless complete subalgebra A such that A 6= B. We prove below that such an algebra exists. The question whether a simple algebra exists was first raised in [8] where it was proved that B has no proper atomless complete subalgebra if and only if B is rigid and minimal. For more on this problem, see [4], [5] and [1, p. 664]. Properties of complete Boolean algebras correspond to properties of generic mod- els obtained by forcing with these algebras. (See [6], pp. 266{270; we also follow [6] for notation and terminology of forcing and generic models.) When in [7] McAloon constructed a generic model with all sets ordinally definable he noted that the corresponding complete Boolean algebra is rigid, i.e. admitting no nontrivial auto- morphisms. In [9] Sacks gave a forcing construction of a real number of minimal degree of constructibility. A complete Boolean algebra B that adjoins a minimal The first author was supported in part by an NSF grant DMS-9401275. The second author was partially supported by the U.S.{Israel Binational Science Foundation. Publication No. 566 Typeset by AMS-TEX 1 Paper Sh:566, version 1996-05-04 10. See https://shelah.logic.at/papers/566/ for possible updates. set (over the ground model) is minimal in the following sense: (1) If A is a complete atomless subalgebra of B then there exists a partition W of 1 such that for every w 2 W , Aw = Bw, where Aw = fa · w : a 2 Ag: In [3], Jensen constructed, by forcing over L, a definable real number of minimal degree. Jensen's construction thus proves that in L there exists rigid minimal complete Boolean algebra. This has been noted in [8] and observed that B is rigid and minimal if and only if it has no proper atomless complete subalgebra. McAloon then asked whether such an algebra can be constructed without the assumption that V = L. In [5] simple complete algebras are studied systematically, giving examples (in L) for all possible cardinalities. In [10] Shelah introduced the (f; g)-bounding property of forcing and in [2] devel- oped a method that modifies Sacks' perfect tree forcing so that while one adjoins a minimal real, there remains enough freedom to control the (f; g)-bounding property. It is this method we use below to prove the following Theorem: Theorem. There is a forcing notion P that adjoins a real number g minimal over V and such that B(P) is rigid. Corollary. There exists a countably generated simple complete Boolean algebra. The forcing notion P consists of finitely branching perfect trees of height !. In order to control the growth of trees T 2 P, we introduce a master tree T such that every T 2 P will be a subtree of T . To define T , we use the following fast growing 1 1 sequences of integers (Pk)k=0 and (Nk)k=0: Pk (2) P0 = N0 = 1;Pk = N0 · ::: · Nk−1;Nk = 2 211 (Hence Nk = 1; 2; 4; 256; 2 ;::: ). Definition. The master tree T and the index function ind: (3)(i) T ⊂ [!]<!, 2 Paper Sh:566, version 1996-05-04 10. See https://shelah.logic.at/papers/566/ for possible updates. (ii) ind is a one-to-one function of T onto !, (iii) ind (< >) = 0, (iv) if s; t 2 T and length(s) < length(t) then ind(s) < ind(t); (v) if s; t 2 T , length(s) = length(t) and s <lex t then ind(s) < ind(t); (vi) if s 2 T and ind(s) = k then s has exactly Nk successors in T , namely all _ s i, i = 0;:::;Nk − 1. The forcing notion P is defined as follows: Definition. P is the set of all subtrees T of T that satisfy the following: (4) for every s 2 T and every m there exists some t 2 T , t ⊃ s, m such that t has at least Pind(t) successors in T . (We remark that T 2 P because for every m there is a K such that for all k ≥ K, m P Pk ≤ 2 k = Nk.) When we need to verify that some T is in P we find it convenient to replace (4) by an equivalent property: Lemma. A tree T ⊆ T satisfies (4) if and only if (5)(i) every s 2 T has at least one successor in T , (ii) for every n, if ind(s) = n and s 2 T then there exists a k such that if n ind(t) = k then t 2 T , t ⊃ s and t has at least Pk successors in T . Proof. To see that (5) is sufficient, let s 2 T and let m be arbitrary. Find some s 2 T such that s ⊃ s and ind(s) ≥ m, and apply (5ii). The forcing notion P is partially ordered by inclusion. A standard forcing ar- gument shows that if G is a generic subset of P then V [G] = V [g] where g is the generic branch, i.e. the unique function g : ! ! ! whose initial segments belong to all T 2 G. We shall prove that the generic branch is minimal over V , and that the complete Boolean algebra B(P) admits no nontrivial automorphisms. First we introduce some notation needed in the proof: (6) For every k, sk is the unique s 2 T such that ind(s) = k. 3 Paper Sh:566, version 1996-05-04 10. See https://shelah.logic.at/papers/566/ for possible updates. (7) If T is a tree then s 2 trunk(T ) if for all t 2 T , either s ⊆ t or t ⊆ s. (8) If T is a tree and a 2 T then (T )a = fs 2 T : s ⊆ a or a ⊆ sg: Note that if T 2 P and a 2 T then (T )a 2 P. We shall use repeatedly the following technique: Lemma. Let T 2 P and, let l be an integer and let U = T \ !l (the lth level of T ). Let x_ be a name for some set in V . For each a 2 U let Ta ⊆ (T )a and xa be such that Ta 2 P and Ta x_ = xa. 0 S 0 0 l l Then T = fTa : a 2 Ug is in P, T ⊆ T , T \ ! = T \ ! = U, and 0 T x_ 2 fxa : a 2 Ug. We shall combine this with fusion, in the form stated below: 1 1 Lemma. Let (Tn)n=0 and (ln)n=0 be such that each Tn is in P, T0 ⊇ T1 ⊇ · · · ⊇ ln ln Tn ⊇ ::: , l0 < l1 < ··· < ln < : : : , Tn+1 \ ! = Tn \ ! , and such that (9) for every n, if sn 2 Tn then there exists some t 2 Tn+1, t ⊃ sn, with n length(t) < ln+1, such that t has at least Pind(t) successors in Tn+1. T1 Then T = n=0 Tn 2 P. Proof. To see that T satisfies (5), note that if sn 2 T then sn 2 Tn, and the node t found by (9) belongs to T . We shall now prove that the generic branch is minimal over V : Lemma. If X 2 V [G] is a set of ordinals, then either X 2 V or g 2 V [X]. Proof. The proof is very much like the proof for Sacks' forcing. Let X_ be a name for X and let T0 2 P force that X_ is not in the ground model. Hence for every T ≤ T0 0 00 0 _ 00 _ there exist T , T ≤ T and an ordinal α such that T α 2 X and T α2 = X. 0 0 Consequently, for any T1 ≤ T and T2 ≤ T there exist T1 ≤ T1 and T2 ≤ T2 and 0 0 _ 0 _ an α such that both T1 and T2 decide \α 2 X" and T1 α 2 X if and only if 0 _ T2 α2 = X. 4 Paper Sh:566, version 1996-05-04 10. See https://shelah.logic.at/papers/566/ for possible updates. 1 1 ln Inductively, we construct (Tn)n=0,(ln)n=0, Un = Tn \ ! , and ordinals α(a; b) for all a; b 2 Un, a 6= b, such that (10)(i) Tn 2 P and T0 ⊇ T1 ⊇ · · · ⊇ Tn ⊇ ::: , (ii) l0 < l1 < ··· < ln < ··· , ln ln (iii) Tn+1 \ ! = Tn \ ! = Un, (iv) for every n, if sn 2 Tn then there exists some t 2 Tn+1, t ⊃ sn, with n length(t) < ln+1, such that t has at least Pind(t) successors in Tn+1, (v) for every n, for all a; b 2 Un; if a 6= b then both (Tn)a and (Tn)b decide _ _ _ \α(a; b) 2 X" and (Tn)a α(a; b) 2 X if and only if (Tn)b α(a; b) 2 X. T1 When such a sequence has been constructed, we let T = n=0 Tn. As (9) is satisfied, we have T 2 P and T ≤ T0. If G is a generic such that T 2 G and if X is _ the G-interpretation of X then the generic branch g is in V [X]: for every n, g ln is the unique a 2 Un with the property that for every b 2 Un, b 6= a, α(a; b) 2 X if _ and only if (T )a α(a; b) 2 X.
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